Construction of the effective action in nonanticommutative supersymmetric field theories

We develop a general gauge invariant construction of the one-loop effective action for supersymmetric gauge field theories formulated in $\mathcal{N}=1\mathrm{/}2$ superspace. Using manifestly covariant techniques (the background superfield method and proper-time representations) adopted to the $\mathcal{N}=1\mathrm{/}2$ superspace we show how to define unambiguously the effective action of a matter multiplet (in fundamental and adjoint representations) and the vector multiplet coupled to a background $\mathcal{N}=1\mathrm{/}2$ gauge superfield. As an application of this construction we exactly calculate the low-energy one-loop effective action of matter multiplet and SU(2) SYM theory on the Abelian background.